Nine geometricall exercises, for young sea-men and others that are studious in mathematicall practices: containing IX particular treatises, whose contents follow in the next pages. All which exercises are geometrically performed, by a line of chords and equal parts, by waies not usually known or practised. Unto which the analogies or proportions are added, whereby they may be applied to the chiliads of logarithms, and canons of artificiall sines and tangents. By William Leybourn, philomath.

PROPOSITIONS GEOGRAPHICALL, Shewing how the Distance of any two Places upon the Terrestriall Globe may be found, both by Trigono∣metricall Calculation and Geometricall Projection. The Eighth EXERCISE.

BEfore I come to shew how the Distances of Places are to be found upon the Terrestri∣all Globe, it will be necessary, first, to de∣scribe unto you the manner how any two Pla∣ces, whose Longitudes and Latitudes are given, may be laid down upon the Projecti∣on. Wherefore in the Scheme, the outward Circle thereof, N AE S ae, represents the first Meridian, passing over the Islands of Azores, from whence the Ancients did begin their account of Longitude, because (say some) the Compass hath there no Variation.—The Line AE ae is the Aequinoctial, upon which the Longitude is counted from the first Meridian.—The Circles N E S, N R S, N C S, and N D S, are Circles of Longitude passing over severall Places.—The lesser Circles E, C, D, and V, are Circles or Parallels of Latitude.

—The Points *, *, *, &c. are severall Places whose distance we are to find by the following Propositions.—And the Points Sun;, ☉, ☉, &c. are the Poles of the Arches of great Circles which pass through the respective Places whose distance is to be found.

And here note, That the Circles of Longitude in this Pro∣jection are the same as the Azimuth Circles in the former Pro∣jection; and the Centres and Poles of them are found in the same manner.—Likewise, the Parallels of Latitude in this Scheme are the same with the Parallels or Circles of Altitude in the former Projection, and their Centres are found in the same manner as is before, in the Description of that Projection, prescribed; and therefore it shall here need no more Precepts for its Delineation, but we will proceed to the Propositions which shew how to find the Distance of Places.

PROP. I. Two Places which differ onely in Latitude, to find their Distance.

IN this Proposition there are two Varieties.

1. If both the Places lie under one and the same Meridian, and on one and the same Side of the Aequinoctial, either on the North or South Side thereof, then substract the lesser Lati∣tude from the greater, and the Difference converred into Miles (by allowing 60 Miles to one Degree) shall give you the Distance.

Example. London and Ribadio lie both under one Meridian, namely of 20 degr. of Longitude; but they differ in Latitude, for London hath 51 d. 30 min. and Ribadio hath Latitude 43 d. both North; the difference of Latitude is 8 degr. 30 m. which being turned into Miles makes 510 miles.

2. If the two Places lie under one and the same Meridian,

but one on the North, and the other on the South-side of the Aequinoctial, adde both the Latitudes together, the Sum is the Distance.

[illustration] geometrical diagram

Example. London and the Island Tristan Dacunhu lie both under one Meridian; but London hath 51 degr. 30 min. North Latitude, and the Island hath 34 d. South Latitude: their Sum is 85 degr. 30 min. which converted into Miles (by dividing

the Degrees by 60. and allowing for every Minute one Mile) makes 5130 miles. And such is the distance of London and the Island Tristan Dacunhu.

To find the distance of these Places upon the Projection.

Seeing that they all lie under one Meridian, namely, N E G H S, find the Pole thereof at K; then lay a Ruler to K and E, it will cut the first Meridian in a; also a Ruler laid from K to G will cut the Meridian in b: the distance a b, measured upon the Line of Chords, will give 8 degr. 30 min. the Distance of London and Ribadio. Again, to find the Distance between London and the Island Tristan Dacunhu, lay a Ruler from K to E, it will cut the first Meridian in a, (as before) and laid from K to H, it will cut the first Meridian in c: the Distance a c, being mea∣sured upon the Line of Chords, will contain 85 degr. 30 m. the Distance between London and the Island, which in Miles is 5130.

PROP. II. Two Places which differ onely in Longitude, to find their Distance.

IN this Proposition there are two Varieties also. For 1. The two Places may lie both under the Aequinoctial, and have no Latitude: in this Case the difference of their Lon∣gitudes (if it be less then 180 degr.) reduced into Miles is their Distance; but if their difference exceed 180 degr. take it out of 360 degr. the remaining Degrees turned into Miles will be the Distance of the two Places.

Example. The Island Sumatra and the Island of S. Thoma lie both under the Aequinoctial, the Island of S. Thoma having 33 d. 10 m. of Longitude, and the Island Sumatra 137 d. 10 m. The lesser Longitude taken from the greater leaves 104 d. 0 m.

which converted into Miles is 6240. And that is the Distance of the two Islands.

[illustration] geometrical diagram

2. But if the two Places differ onely in Longitude, and lie not under the Aequinoctial, but under some other intermediate Parallel of Latitude, between the Aequinoctial and one of the Poles, then to find their Distance, this is

The Analogie or Proportion.

As the Radius 90 degr. is to the Co-sine of the common Lati∣tude 47 degr.

So is the Sine of half the difference of Longitude 21 d. 37 m. to the Sine of half their Distance 15 degr. 38 m.

So let the two Places be the Cities Constantinople and Com∣postella, both lying in the Latitude of 43 degr. but they differ in Longitude 43 degr. 15 min. So that their Distance accor∣ding to the former Analogie will be found to be 31 degr. 16 m. which, converted into Miles, is 1876 miles, which is the Di∣stance between the two Cities Constantinople and Compostella.

To find the Distance of these two Places upon the Projection.

The two Places upon the Projection are noted with the Letters C and D, both lying in the Latitude of 43 degr. but Constantinople in the Longitude of 63 degr. and Compo∣stella in the Longitude of 106 d. 15 min. So that their diffe∣rence of Longitude is 43 degr. 15 min. Wherefore through the two Places C and D draw the Arch of a great Circle, and find the Pole thereof; (which to effect is already taught at the beginning of this Book:) which Pole will be at the Point M. Then laying a Ruler upon M and C, it will cut the first Meridian in the Point d; and laid from M to D, it will cut the first Meridian in e: the Distance between d and e, mea∣sured upon the Line of Chords, will be found to contain 31 degr. 16 min. which, converted into Miles, giveth 1876, the Distance as before.

PROP. III. Two Places differing both in Longitude and Latitude being proposed, to find their Distance.

THERE are three Varieties contained in this Propositi∣on. For

1. One of the Places may lie under the Aequinoctial, and have no Latitude; and the other under some Parallel of La∣titude, between the Aequinoctial and one of the Poles. For finding the Distance of Places that are so situate, this is

The Analogie or Proportion.

As the Radius 90 degr. is to the Co-sine of the difference of Longitude 76 degr. 50 min.

So is the Co-sine of the Latitude given 38 degr. 30 min. to the Co-sine of the Distance required 52 degr. 41 min.

Thus suppose the two Places to be the Island of S. Thoma, ly∣ing under the Aequinoctial in the Longitude of 33 d. 10 m. and London under the Parallel of 51 d. 30 m. of North Latitude, having 20 degr. of Longitude, their Distance by the former Proportion will be found to be 52 degr. 41 min. which, con∣verted into Miles, gives 3161 miles for their Distance.

By the Projection.

The two Places upon the Projection are represented by the Letters A and E. The Point A lying under the Aequinoctial, and in 33 degr. 10 min. of Longitude, represents the Island of S. Thoma; and the Point E, under the Parallel of 51 d. 30 m. North, and in Longitude 20 degr. represents London.— Through the two Places A and E (according to former Dire∣ctions) draw an Arch of a great Circle, and find the Pole thereof, which will be at the Point L. A Ruler laid to L and

the Point E will cut the first Meridian in n; and the Ruler being laid from L to A will cut the first Meridian in f: the Di∣stance n f, being measured upon the Line of Chords, will be found to contain 52 degr. 41 min. as before, which in Miles is 3161.

[illustration] geometrical diagram

2. If both the Places proposed shall be without the Aequi∣noctial, but on one Side, either both towards the North, or

both towards the South, the finding of their Distance is by this

Analogie or Proportion.

(1.) As the Radius 90 degr. is to the Co-sine of the diffe∣rence of Longitude 44 degr.

So is the Tangent of 38 degr. 30 min. to a fourth Tangent 28 degr. 55 min. which taken from the Complement of the Lat. of Jerusalem 58 degr. 20 min. leaves 29 d. 25 m.

(2.) As the Co-sine of the fourth Tangent 61 degr. 35 min. is to the Co-sine of 60 degr. 35 min.

So is the Co-sine of the Latitude of London 38 degr. 30 min. to the Co-sine of the Distance 51. degr. 9 min.

So the two Places propounded being London, lying in North Latitude 51 degr. 30 min. and Longitude 20 degr. and the other, Jerusalem, lying in North Latitude also 31 degr. 40 min. and Longitude 66 degr. you may find their Distance by the foregoing Analogie to be 38 degr. 51 min. which in Miles makes 2331.

By the Projection.

The two Places in the Projection are represented by the Letters E and F, E being London, F Jerusalem; through which Points draw the Arch of a great Circle, and find its Pole: the Circle (in this Example) comes so near a right Line, that I have so drawn it; and therefore his Pole is but little within the outward Circle, viz. at P. Wherefore lay a Ruler to P and E, it will cut the first Meridian in g; and being laid from P to F, it will cut the Meridian in h: the Distance g h, being measured upon the Line of Chords, will be found to con∣tain 38 degr. 51 min. and in Miles 2331, as before.

3. The two Places propounded may be so situate, that one of them may lie on the North, and the other on the South-side of the Aequinoctial. For finding the Distance of such Places follow this

Analogie or Proportion.

(1.) As the Radius 90 degr. is to the Co-sine of the difference of Longitude 40 degr.

So is the Co-tangent of the greater Latitude 50 d. to the Tan∣gent of a fourth Arch 37 d. 10 m. which being substracted out of the other Latitude, and 90 d. added thereto, say,

[illustration] geometrical diagram

(2.) As the Co-sine of the Arch found 52 degr. 50 min. is to the Co-sine of the Arch remaining 52 degr. 50 min.

So is the Co-sine of the Latitude first taken 50 degr. 00 min. to the Co-sine of the Distance 40 degr. which taken from 180 degr. there remains 140 degr. for the Distance of the two Places.

So the two Places propounded being the Cape of Good hope, lying in the Latitude of 40 degr. South, and Longitude 50 d. and the other Place Malibrigo, lying in 26 degr. of North La∣titude, and in 180 degr. of Longitude, you may find their Di∣stance by the foregoing Analogie to be 140 degr. which, con∣verted into Miles, make 8400. And such is the Distance of the two Places.

By the Projection.

In the Projection the two Places are represented by the Let∣ters T and V; the Letter V representing the Cape of Good hope, and T Malibrigo. Now Malibrigo having 180 degr. of Longi∣tude, (which is just half the Circumference of the Aequinoctial, and is as far remote as any Place can be from the first Meridi∣an; for if you were to project any Place having above 180 d. Longitude, (as suppose 230 degr.) you must substract such Longitude from 360 degr. and project the remainer; so 230 degr. being taken from 360 degr. leaves 130 degr. which must be projected in stead of 230 degr. and by this means it is that Malibrigo is projected upon the outermost Circle or first Meridian.)

Through these two Points T and V draw the Arch of a great Circle, T V X, and find its Pole at R: then a Ruler laid at R and the Point V will cut the first Meridian in k, and T k, being measured upon your Line of Chords, will be found to contain 140 d. and that is their Distance, which in Miles maketh 8400.

These are all the Varieties of Positions of Places upon the Terrestriall Globe; for no two Places (whose Distance can be

required) can be proposed, but they must fall under one or other of the Varieties contained in some of these three Propositions. And note that this way of finding the Distance of Places is the most absolute and exact of any other.—And what is here said con∣cerning finding these Distances the ingenious may apply to Cir∣cular Sailing, of all other waies the most perfect: which I shall leave to the industrious Sea-man to find out of himself, till I pre∣sent him with something of that kind: in the mean time let him make use of the foregoing EXERCISES, and this which follows.